News From the Front, XI: Holographic Heat Engines!

Yes, you heard me right. Holographic Heat Engines. I was thinking recently about black holes in universes with a cosmological constant and their thermodynamics. I had an idea, it led to another, then another, then some calculations, and then a couple of days of writing, calculating, and thinking… then a day to cool off and think about other things. Then I came back to it, decided it was still exciting as an idea and so tidied it all up as a paper, made some diagrams, tidied some more, and voila! A paper submitted to the arXiv.

I’m sort of pleased with all of it since it allowed me to combine a subject I think is really fun (although often so bleakly dull when presented at undergraduate level) – heat engines – with contemporary research ideas in quantum gravity and high energy physics. So I get to draw some of the cycles in the p-V plane (graph of pressure vs volume) representing the inner workings of engines of particular designs (just like you might have seen long ago in a physics class yourself) and compute their efficiency for doing mechanical work in exchange for some heat you supply. It is fundamental that you can’t do that with 100% efficiency otherwise you’d violate the second law of thermodynamics – that’s why all engines have to have some exhaust in the form of heat, giving an efficiency represented by a quantity [tex]\eta[/tex] that is less than one, where one is 100% efficient. The diagram on the left illustrates the key pieces all engines must have, no matter what working substance you’re using. The details of the design of the engine are what kind of cycle you taking it through and what the properties (“equation of state”) your working substance has. In the case of a car, for example, the working substance is cleverly mixed up with the source of heat – the air/gasoline mix forms a “working substance” that gets expanded and compressed in various ways (in the green bit of the diagram), but the fact that it also burns releasing heat means it is also the source of the heat that comes into the engine (the flow from the red bit) to be (in part) turned to work, and the remainder flowing out to the blue (exhaust). Very clever.

The cool thing here is that I’m using black holes as the working substance for my engines! You ask: Is this why you painted the engine green, cvj? You’ve found the ultimate green energy source and in a future where all our clothes are tight-fitting and weirdly short at the cuff and ankle we’re all going to fill up at the pump with premium grade black holes…? Thanks for asking that. No. First of all, the black holes are the working substance, not the source of the heat, and moreover, well… you can imagine the other engineering difficulties!

Let’s take a step back. This is an interesting exercise for a number of reasons, but mostly to do with understanding properties of quantum field theories, to my mind. Although I suspect that these are going to be useful concepts to play with in ways I have not yet thought of. A fun aspect of all of this is that nobody seems to have done this before.

But people have thought about extracting energy from black holes for many years now, right? How can you say that this has not been done before? Also excellent questions. Well, that’s totally different. Indeed, astrophysicists need to know about mechanisms for extracting energy from rotating black holes since they are the sources of some of the most spectacular events in our universe, such as active galactic nuclei, powerful x-ray jets, and so on and so forth. There are very many excellent papers on those matters, starting with the observation of Penrose (the Penrose process) and going to detailed mechanisms like the Blandford-Znajek process. All really interesting, but those are not heat engines – they are not extracting mechanical work from heat like your (or a friend’s) car does. People have also considered making engines of other kinds, getting black holes to do things involving spinning up or down, or moving charges in potentials, and so forth. But this is also not what I’m talking about. The point there is that those other processes have what are called “work terms” in the first law of thermodynamics that was written down for black holes back in the 1970s starting with work of Bekenstein, and then completed by Hawking and others. But until recently the most standard work term, -pdV, taking about direct mechanical work, has not been present in the black hole thermodynamics at all. That’s the key.

This all began to change in recent times when people began to study the possibility of studying black hole thermodynamics but letting the cosmological constant (vacuum energy) vary as a variable too. After all, it is well known that a cosmological constant is in fact equivalent to a pressure (p) associated with spacetime. Up to factors, one is minus the other. Our universe might have a small positive cosmological constant for example, which means a negative pressure. That corresponds to having spacetime wanting to expand if left to its own devices. Just the sort of expansion that was observed in 1998 (and was the subject of 2011’s Nobel Prize for Physics) in the form of the accelerating expansion of the universe. Dark energy behaves like a cosmological constant. Positive pressure comes from negative cosmological constant – the kind we are much better at handling in quantum gravity research so far, in the context of string theory and AdS/CFT and its extensions, which you’ve maybe heard a lot about here and elsewhere.

Also, positive pressure fits better with intuition about how engines work thermodynamically, so that’s neat for my idea. What’s the idea? Well, as I say in the paper, as soon as pressure and volume (a more subtle quantity here) are in play in the laws of black hole thermodynamics (as a result of the hard work of a lot of people who I cite at the beginning of the paper), it would seem neglectful to not consider making traditional heat engines, albeit with a highly non-traditional working substance. So I did my duty and considered it, and that’s what the paper is partly about. I make such heat engines (er., on paper).

For the case of non-rotating black holes, the full equation of state can be written explicitly and so one can play with making engines of different kinds and computing their efficiency. But the Big Kajuna in all of this is the Carnot engine. It has the most efficient any engine can have given the limiting temperatures of the hot and cold heat exchanges that happen. [tex] \eta=1-T_C/T_H\ .[/tex] Would I be able to talk about it in principle only, or be extremely explicit? Turns out that static black holes are very special, and it allows for an explicit characterisation of a static black hole Carnot cycle. In fact, for those of you still reading and who know the language, it is equivalent to a black hole Stirling cycle because isochoric paths and adiabatic paths coincide in this case!

But wait! you cry. What are we supposed to do with these black hole heat engines now we know they can exist? How do we even operate them? Aha! Well, I’m sure there will be lots to do with these things that I’ve not worked out fully yet, and I’m sure others will come up with their own ideas. But the details require figuring out exactly what this mechanical work actually is, and holography (mapping all the physics to a system in one dimension fewer without gravity) helps interpret it all and give clues for how to operate the engines. It required me to tackle something else that had not been worked out in the literature before, as far as I can tell. What is the meaning of this extended black hole thermodynamics (containing a p and a V) in the holographic dual field theory? Does it replace the usual traditional black hole thermodynamics and how it fits with holography, or is it just not relevant? In the paper I propose just how the traditional AdS/CFT holography extends to make sense of these questions. The clue is that the pressure and the volume of the gravity theory are not pressure and volume of the dual theory. They have a different meaning altogether. In the end, both set of thermodynamics are true, when interpreted correctly.